I only experience this minor issue when the car is cold. I crank and it idles fine but when I put it in gear and start to move it starts stumbling but not to the point where it's gonna stall, just about to 500rpm (idle afaik is 650) it only happens when I first move it, doesn't happen after that until I have another cold start (sorry if it doesn't make that much sense)

I think I have a vacuum leak coming from one of the intake gaskets because they're old and have never been changed. What else could it be?

Can you replicate this behaviour while at idle, standing over your engine and pressing the throttle cable ?

Then spray carb cleaner at all intake and vacuum ports and see and redo the test to see if the engine behaves better.

Inspect everything in your intake that you can get your hands on without major disassembly, such are the main air hose. Make sure connections are tightened.

I doubt rubber gaskets expand when heated. They are not made of metal.

Your cold start problems could well be a defective crankshaft sensor. The resistance when cold could be falling outside its rated specs. Check the resistances, when cold, and when hot, and anyway, you know the drill sir, if you intend to keep the car and haven't changed it out in the past few years, do so to avoid the inevitable no-start situation.